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binomial(i,j)*11^(i-j)*8^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:             1;\n  1:            27,             8;\n  2:           665,           432,            64;\n  3:         15523,         15960,          5184,           512;\n  4:        349809,        496736,        255360,         55296,          4096;\n  5:       7688939,      13992360,       9934720,       3404800,        552960,         32768;\n  6:     165898633,     369069072,     335816640,     158955520,      40857600,       5308416,       262144;\n  7:    3528441075,    9290323448,   10333934016,    6268577280,    2225377280,     457605120,     49545216,     2097152;\n  8:   74191354337,  225820228800,  297290350336,  220457259008,  100297236480,   28484829184,   4881121280,   452984832,   16777216;\n  9: 1545504236731, 5341777512264, 8129528236800, 7134968408064, 3968230662144, 1444280205312, 341817950208, 50205818880, 4076863488, 134217728;\n","recurrence_sentence":"The row polynomials satisfy (8*n+11+8*x)*P(n,x) = (152*n+361+(64*n+304)*x+64*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((152*n + 361 + (64*n + 304)*x + 64*x^2)*P[n - 1, x])/(8*n + 11 + 8*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((152*n + 361 + (64*n + 304)*x + 64*x^2)*v[i - 1])/(8*n + 11 + 8*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, 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